This study builds upon a cooperative game framework based on the Cartesian product of two sets, as introduced by M. Slime et al., by proposing a more flexible participation model. In one of their previous studies published in 2024, the authors assumed that all players must participate simultaneously in both games. Here, we relax this constraint by allowing each player to independently decide whether to participate in one, both, or neither of the games. This generalization better reflects real-world situations in multi-agent systems, such as service provision agreements involving agents with differing preferences or capabilities.

We redefine coalition structures and the associated payoff functions within this extended framework and investigate how classical solution concepts—specifically the core and the Shapley value—can be adapted to this setting. While the relaxation introduces additional complexity to the analysis, it enables a richer and more inclusive model of partial cooperation. An illustrative example is provided, and conditions are identified under which desirable properties are achieved.